The remarkable warped and twisted gas disk in NGC 3718
Linda S. Sparke, Gustaaf van Moorsel, Ulrich J. Schwarz, Martin Vogelaar
TThe remarkable warped and twisted gas disk in NGC 3718
Linda S. Sparke
University of Wisconsin, Department of Astronomy, Madison WI 53706 [email protected]
Gustaaf van Moorsel
P. O. Box 0, National Radio Astronomy Observatory, Socorro, NM 87801 [email protected]
Ulrich J. Schwarz
Department of Astrophysics, Radboud University, PO Box 9010, NL-6500 GL Nijmegen andKapteyn Astronomical Institute, PO Box 800, 9700 AV Groningen, the Netherlands [email protected]
Martin Vogelaar
Kapteyn Astronomical Institute, PO Box 800, 9700 AV Groningen, the Netherlands [email protected]
ABSTRACT
We have mapped NGC 3718, a nearby bright galaxy in a loose group, andits companion NGC 3729 in the 21 cm line of neutral hydrogen. NGC 3718 isa strikingly unusual galaxy with a strong straight dust lane across the center,peculiar diffuse spiral arms, and an extended disk of neutral hydrogen. Earlierwork showed the gas disk to be strongly twisted, warping through edge-on wherewe see the straight dust lane; stars formed in this gas appear to make up the ‘spiralarms’. Our improved maps show a twisted but bisymmetric disk of gas extendingto 7 (cid:48) or 35 kpc, where the orbital period is roughly 1 Gyr. It is surrounded byfragmentary spiral features, and a streamer of gas extending to a cloud lying 12 (cid:48) or 60 kpc to the north. We use inspector , a task in gipsy , to fit a tilted-ringmodel interactively to slices through the H i data cube. The apparent major axisswings through 100 ◦ from the innermost gas orbits at 30 (cid:48)(cid:48) radius to the outer edge.When viewed in the reference frame of the galaxy’s stellar disk, the innermost a r X i v : . [ a s t r o - ph . GA ] F e b ◦ –40 ◦ .The line of nodes, where the gas orbits pass through the plane of the stellar disk,twists by roughly 90 ◦ about the pole. We do not see gas orbiting in the planeof the stellar disk. If we assume that the galaxy’s dark halo shares the samemidplane, then the observed twist can be explained by differential precession ina dynamical model in which the dark halo is fairly round. The run of tilt withradius is close to what is required for the warped gas disk to precess rigidly inthe galaxy’s gravitational field without changing its shape. This fact probablyaccounts for the longevity of the twisted structure. Subject headings: galaxies: individual (NGC 3718) — galaxies: kinematics anddynamicsFacilities: VLA, WIYN, ADS, NED
1. Introduction
The luminous galaxy NGC 3718 (UGC 6524; Arp 214; PRC D-18) and its dwarfcompanion NGC 3729 form a galaxy pair in the loose Ursa Major group (Tully et al. et al. ◦ into an ‘S’ shape, forming a diffuse spiral in thestellar light which led de Vaucouleurs et al. (1991) to classify the galaxy as a peculiarbarred spiral. As in Centaurus A, an active nucleus is largely hidden behind the dust lane.A compact radio continuum source less than 0.2 pc across has a brightness temperature inexcess of 10 K (Nagar et al. et al. (2007) see a jet stretching 0.5 (cid:48)(cid:48) or 40 pc tothe north-west. Optical spectroscopy shows a LINER of Type 1.9 (Ho et al. α and a strong narrower line of [O i ] at 6300˚A.The H i maps of Schwarz (1985) showed that the dusty gas does not lie in the plane of thestellar disk, but forms a complex three-dimensional structure. Some lines of sight pass morethan once through the gas layer, giving rise to multiple velocity peaks; but the velocityfield is strikingly bisymmetric. Schwarz was able to describe the gas layer as a violentlywarped disk made up of material following concentric but tilted orbits, which twisted byroughly 90 ◦ between the inner and outer radii. The strong straight portion of the dust lanearises where the orbits turn nearly edge-on to our line of sight, at roughly 200 (cid:48)(cid:48) radius.Pott et al. (2004) mapped molecular gas in the inner part of the dust lane using the CO 3 –and HCN lines. Krips et al. (2005) combined those data with interferometric maps at ∼ (cid:48)(cid:48) resolution in the CO 1 → i gas. NGC 3718 isincluded as ‘related object’ D-18 in the Polar Ring Catalog of Whitmore et al. (1990).Because polar ring systems contain gas orbiting in more than one plane, these rare objectsconstitute one of the few observational probes of the three-dimensional mass distribution ofgalaxies. Sparke (1990) presented a dynamical model for NGC 3718 to explain the complexshape of the twisted H i disk mapped by Schwarz (1985). According to this model we seethe underlying disk galaxy almost face-on, with the ring gas in near-polar orbit about it.The tilted gas orbits precess about the symmetry axis of the flattened central galaxy andits dark halo. Orbits at smaller radius precess more rapidly in the galaxy’s gravitationalfield, so the gas disk becomes twisted. This dynamical model reproduced the main featuresof Schwarz’s tilted-ring fit. It was consistent with a spherical dark halo, and indicated anage for the gas disk of 3 − i and CO, using optical images to resolve ambiguity about where the gas lies in frontof the stellar body. In Sections 5 and 6 these are interpreted as showing a near-polar diskof gas that has become twisted by the differential precession of the gas orbits in thegalaxy’s aspherical gravitational potential.Table 1 gives basic information on NGC 3718 and NGC 3729. We adopt the distance of17 Mpc given by Tully (1998) for the Ursa Major group: there, 1 (cid:48) is equivalent to 4.945 kpc,1 (cid:48)(cid:48) = 82.4 pc, and 13 (cid:48)(cid:48) = 1.07 kpc. NGC 3718 is a very luminous galaxy with L B ≈ × L (cid:12) , while for NGC 3729 L B ≈ × L (cid:12) . Tully et al. (1996) classifiedNGC 3718 as T=1 (Sa) and NGC 3729 as T=2 (Sab) on the basis of their optical andnear-infrared images.
2. Observations in the 21cm Line and Data Reduction
We used the VLA in the C configuration in four different observing runs in March andApril 1992, for a total observing time of 26 hours and 28 minutes on source. To obtain therequired velocity coverage and resolution, we observed using two IFs, tuned at slightly 4 –Fig. 1.— Left, B-band image of NGC 3718 taken by E. Wehner with the WIYN 0.9-mtelescope, showing the strong central dust lane and diffuse ‘spiral arms’. North is up andEast is to the left; the image covers 9 (cid:48) × . (cid:48) . Right, R -band image of the central regiontaken by J. Gallagher with the 3.5-m WIYN telescope; the dust lane is close to edge-on,with the bright nucleus seen to the north. The dark feature seen closest to the nucleus is atPA ≈ ◦ . 5 –different frequencies in order to almost double the spectral coverage. Parameters of theobservations under proposal AS649 are listed in Table 2. Compared to the observations ofSchwarz (1985), we improve the velocity resolution to 5 km s − from 33 km s − , and thespatial resolution from a 25 (cid:48)(cid:48) × (cid:48)(cid:48) beam to a 13 (cid:48)(cid:48) circular beam.The complete data reduction was done using the Astronomical Image Processing System(AIPS). The four data sets were calibrated independently, both for amplitude and phasegains errors that vary with time, and for those that vary with frequency. The absolute fluxscale was determined by observing 3C286, which has a well-known flux density. After this,the four databases were combined into one. Inspection of a first mapping of the resultallowed us to determine line-free channels at both edges of the band. The average of thesechannels was subtracted from the uv data set using the AIPS task UVLIN. This new datacube, which now contains just line emission, was used throughout in all subsequentmapping and cleaning.The continuum map shows point sources at the central locations of both NGC 3718 andNGC 3729. Table 3 lists positions and fluxes of these point sources, with uncertainties 0.5 (cid:48)(cid:48) and 1.0 mJy respectively. The sources coincide to within this accuracy with the positionsgiven by Verheijen & Sancisi (2001) at 20 cm and by Krips et al. (2005) at 3 mm; we takethem to represent the center of each galaxy. The presence of a dust lane prevents anaccurate optical position for NGC 3718, but the various values in the literature agree withthis radio position within the error margins. In their Appendix B, Verheijen & Sancisi(2001) quote 11 . ± . ± . (cid:48)(cid:48) beam. A proper choice of robustness (Briggs1995; Briggs et al. − velocity resolution, and once applyinga smoothing over the frequency axis resulting in 10 km s − velocity resolution. The mapswith 10 km s − resolution showed no additional structure in the outer galaxy, so the5 km s − resolution data were used throughout in this paper. Both cubes were cleaned to a1 σ noise level: 0.39 mJy/beam for the full resolution data and 0.30 mJy/beam for thefrequency-smoothed data, as given in Table 2. 6 –
3. Neutral Hydrogen Datacube for NGC 3718 and NGC 37293.1. Global Results
We list global H i properties for NGC 3718 and for NGC 3729 in Table 4. For both galaxies,we used a method which corrects for the mismatch between the dirty beam and the cleanbeam in the residual map (J¨ors¨ater & Van Moorsel 1995) to determine the H i flux in eachchannel map, and corrected for the attenuation of the VLA primary beam. The global H i profiles in Figure 2 show no sign of absorption against the weak radio continuum source ineither galaxy.For NGC 3718, the H i flux integral of 118 Jy km s − agrees with that found by Schwarz(1985) and is 20% lower than the value of Verheijen & Sancisi (2001). Our flux integral of3.8 Jy km s − for NGC 3729 agrees with the latter authors, but is roughly 5 times lowerthan given by Schwarz (1985). A recalculation using the map in Figure 3 of that paperyields a much smaller flux integral, so the result quoted in Schwarz (1985) may have beenin error. Estimates from single-dish observations vary between 90 Jy km s − and150 Jy km s − (Huchtmeier & Richter 1989). Thus there is little gas in an extendedcomponent that would be missed in our maps.In NGC 3718 we find 8 × M (cid:12) of H i gas, about twice as much as in the Milky Way, whilethe galaxy is about 50% brighter in stellar light. The ratio M HI /L B =0.3, which is aboutaverage for the sample of gas-rich S0 and Sa galaxies studied by Noordermeer et al. (2005).NGC 3729 has only 3 × M (cid:12) of H i and is gas-poor compared to a normal Sab galaxy; wefind M HI /L B =0.04 while M HI /L B =0.1 would be typical (Roberts & Haynes 1994).The regular, steep-sided and symmetric profile of NGC 3718 suggests that the gas has hadtime to settle into a steady state. Between the points at which the emission falls to 20% ofits peak value we measure a width W =476 km s − . If the H i followed pure circular orbits,our measured line width would yield the rotation speed directly: W = 2 V max sin i , where V max is the maximum rotation speed in the galaxy disk, inclined at angle i to face-on.Verheijen & Sancisi (2001) find that in disk galaxies we must subtract about 20 km s − from W to correct for random motions in the gas. For NGC 3718 this would imply V max sin i ≈
230 km s − , with little gas in regions where the circular speed is higher.Based on the mean of the velocities at 20% of peak flux, we adopt the systemic velocity V sys =995 km s − for NGC 3718 and V sys =1063 km s − for NGC 3729. For NGC 3718,Verheijen & Sancisi (2001) derived 993 km s − from the midpoint of their H i global profile,and 990 km s − by examining the position-velocity diagram along PA=195 ◦ . For NGC 3729the agreement is even closer: Verheijen & Sancisi (2001) find V sys =1060 km s − and 7 –Fig. 2.— Global profiles of H i emission in both NGC 3718 (solid line) and NGC 3729 (dashedline). Fluxes have been corrected for primary beam attenuation. 8 –1063 km s − for the two methods respectively. Figures 3 and 4 show the channel maps for gas in NGC 3718. In a warped disk made up ofgas on concentric but tilted circular orbits, gas at each velocity above the systemic velocity V sys should have a counterpart at the same interval below V sys , at a position point-reflectedabout the galaxy center. In Figure 5, emission from gas in two extreme channels centred at765 km s − and 1222 km s − has been superposed to show this symmetry.The extreme channel maps containing H i emission are at 755 km s − and 1232 km s − ,separated by almost exactly our measured width W = 476 km s − . Channel maps at5 km s − lower and higher velocity are empty. The emission in these channels extends from30 (cid:48)(cid:48) from the center to 400 (cid:48)(cid:48) , so V rot sin i should be between 230 km s − and 240 km s − overthis entire radial range. Within 30 (cid:48)(cid:48) , either H i gas is largely absent, or V rot sin i isconsiderably lower.Within 300 (cid:48)(cid:48) of the center the band of emission in Figure 5 is narrow, suggesting that wesee the gas orbits within 10 ◦ to 20 ◦ of edge-on. Tracing the ridge line of the emission in theextreme channels then gives us the position angle of the gas orbits, as plotted in Figure 9.The kinematic major axis swings from close to PA=100 ◦ at 30 (cid:48)(cid:48) from the center toPA=190 ◦ at radius 400 (cid:48)(cid:48) . The cube of data can be viewed as a rectangular array of velocity (or frequency) profiles. Itis standard practice to reduce the spectral line data further by forming maps containingthe value of the various moments of each profile. The zeroth moment is a map showing thespatial distribution of total hydrogen; all velocity information is lost. In calculating thezeroth moments we restrict ourselves to that part of the profile where emission is present;this avoids contamination of the total HI map with noise. Our method of separatingemission and noise is automatic: we convolved the cube to a 40 (cid:48)(cid:48) × (cid:48)(cid:48) beam, and masked(blanked) all pixels in the high resolution cube which were below a 3 σ noise level in the low resolution cube. The moment maps are constructed from the unblanked pixels only. Thismethod avoids the addition of unrelated noise to the total HI map, and at the same timemisses little of the low-level HI emission.Figure 6 shows the resulting map. Within 100 (cid:48)(cid:48) of the center, we see a narrow dense ridge 9 –Fig. 3.— Channel maps for the H i distribution in NGC 3718 at intervals of roughly 20 km s − .Our adopted systemic velocity V sys =995 km s − lies midway between the last channel mapin this figure and the first in Figure 4. 10 –Fig. 4.— Continued from Fig. 3: channel maps for the H i distribution in NGC 3718. 11 –Fig. 5.— Superposed channel maps for the H i gas in NGC 3718 at velocities displacedroughly 230 km s − on either side of the systemic velocity. Lower contours, in red, show gascentred at 1221.8 km s − ; upper contours, in blue, show gas centred at 765.3 km s − . 12 –Fig. 6.— Total hydrogen in NGC 3718 and NGC 3729, corrected for the primary beamattenuation. The lowest contour level is at 3 . × atoms cm − or 0.26M (cid:12) pc − , approxi-mately at the 3- σ noise level. Higher contours are at 1.0, 3.3, 10.0, and 33.4 10 atoms cm − .The positions of the central continuum sources are marked with crosses; the beam size isindicated by the small circle in the lower left hand corner. 13 –of H i emission along PA=140 ◦ , coinciding with the dark dust lane of Figure 1. At largerradii this ridge swings counter-clockwise into a ‘S’ shape. The left panel of Figure 1 showsthat the diffuse ‘spiral arms’ that we see in the starlight correspond to regions where the H i density in Figure 6 rises above 3 . × atoms cm − or 2.6M (cid:12) pc − . This gas densitybarely reaches the threshold of 3–10M (cid:12) pc − normally required for widespread starformation in a galaxy disk ( e.g. Schaye 2004). The tilted-ring model developed for the H i layer in Section 4 below, and illustrated in Figure 7, implies that the projected densityalong the ‘spiral arms’ is increased by warping in the gas layer. The true surface density iseven further below the normal threshold for star formation.The stellar ‘spiral arms’ extend to roughly 250 (cid:48)(cid:48) , while the pattern of H i emission isbisymmetric to about 7 (cid:48) or 35 kpc from the center, and the H i disk can be traced to aradius of 500 (cid:48)(cid:48) or 41 kpc. To the southeast a spiral-arm fragment extends to a longstreamer, apparently ending in a gas cloud projected 12 (cid:48) or 59 kpc from the center. Thisarm fragment and a symmetrically placed structure to the northeast are also visible in theleft panel of Figure 1 as star-forming regions. As in NGC 1058 and NGC 6946 (Ferguson et al. et al. − velocity resolution further to a 40 (cid:48)(cid:48) beam yields anoise level of 0.5mJy/beam. A map of total H i made from this smoothed cube fails to showemission more extended than Figure 6, to a surface density of 0.1M (cid:12) pc − or1 . × atoms cm − . In particular, there is no bridge of emission linking NGC 3718 withNGC 3729.Beyond 200 (cid:48)(cid:48) from the center, we see only a single velocity peak along each line of sight,and the velocity dispersion is generally below 10 km s − . Here the first-moment map ofFigure 7 describes the velocity field of the gas. It shows a pattern characteristic of awarped rotating disk: the kinematic major axis (where the velocity is furthest fromsystemic) twists with radius into an ‘S’-shape. The orderly rotation and low velocitydispersion suggests that the structure is at least a few orbits old. Beyond about 7 (cid:48) or35 kpc, the gas of the spiral-arm fragments seen on both sides of the disk along PA=120 ◦ appears to share in the rotation, although it does not form part of a complete ring. Thelong streamer curving northwards away from the east side of the disk is continuous in bothposition and velocity. Taking the maximum radius as 500 (cid:48)(cid:48) and the circular speed there as220 km s − (see below) yields a dynamical mass M dyn = 5 × M (cid:12) , so that M dyn /L K ≈ et al. i emission in NGC 3718 andNGC 3729; the tilted ring model of Section 4 for the warped gas layer of NGC 3718 issuperposed. Contours are spaced at intervals of 50 km s − around the systemic velocityof 995 km s − . All the contours in the gas streamer on the northeast side of the disk ofNGC 3718 are at 795 km s − . Beyond about 200 (cid:48)(cid:48) from the center, where velocity profiles aresingly peaked, the values in this map are representative of the radial velocity of the gas atthat position. The beam size is indicated by the small circle in the lower left hand corner. 15 –The small galaxy NGC 3729 is projected 11 (cid:48) to the east, and shows a clear signature ofrotation in gas that extends to 1 (cid:48) or 5 kpc radius. ¿From their K-band images, Tully et al. (1996) find an isophotal ellipticity e = 1 − b/a = 0 .
32. If we assume the disk to be round(although the galaxy is classified as barred), it is inclined 48 ◦ from face-on. Furtherassuming that the H i gas shares this plane yields a dynamical mass M dyn = 35 × M (cid:12) and M/L K ≈ ◦ , along the ridge of bright HI emission visible in Figure 6. This plot is highlysymmetrical, as expected for gas in circular orbit about the galaxy center. There are twomain components: the very strong inner one shows velocities rising steeply to 230 km s − at80 (cid:48)(cid:48) from the center, while in the outer component rotation speeds increase almost linearlyto 60 km s − at 300 (cid:48)(cid:48) radius. We interpret the slower-rotating gas as following an orbit atlarger radius; we see only the portion projected close to the center, where the radialvelocity is small. On the western side and at negative velocities, there is a third and muchweaker component with an intermediate slope. Looking towards any point along this linewithin 80 (cid:48)(cid:48) of the center, we would see a double or triple peak in the H i velocity profile. Ifthe H i gas forms a continuously warped disk and we look once through it in the outerparts, then each line of sight must cross the gas layer an odd number of times, so we expecttriple profiles. There is a slight indication that the weak third component may have acounterpart to the east at positive velocities.Krips et al. (2005) made interferometric maps of the molecular gas associated with theinner part of the dust lane. They combined several pointings with the single-dishobservations of Pott et al. (2004) to probe the dust lane to 70 (cid:48)(cid:48) radius with ∼ (cid:48)(cid:48) resolutionin the CO 1 → ◦ . Like our Figure 8, it is highly symmetric about the center, withemission at velocities rising to 220 km s − at 70 (cid:48)(cid:48) radius, and an even smaller-scale structurewhere speeds reach 250 km s − within 10 (cid:48)(cid:48) of the center. This nuclear component wouldcorrespond to an edge-on disk of diameter 1.5 kpc. If atomic and molecular gas share thesame kinematics, then H i must be largely absent within 30 (cid:48)(cid:48) of the center; otherwise, ourFigure 5 would not show a gap in high-velocity emission close to the center. Reshetnikov &Combes (1994) measured velocities in the H α line of ionized gas in the central regions ofNGC 3718. Along PA=130 ◦ , they found velocities rising to 260 ±
20 km s − within 80 (cid:48)(cid:48) ofthe center, which is consistent with the results in CO. 16 –Fig. 8.— Position-velocity plot through the center of NGC 3718, and along the apparent HIridge at PA=140 ◦ . The axes are labeled relative to the systemic velocity V sys =995 km s − and the center of NGC 3718: right ascension increases to the left. The lowest contour is at0.75 mJy/beam; higher contours are odd multiples. The sense of the velocity axis is chosenfor comparison with the figures of Krips et al. (2005). 17 –
4. Tilted-ring models for the H i gas4.1. Fitting a tilted-ring model Because of the very high degree of symmetry in the channel maps of Figures 3 and 4, wefollow Schwarz (1985) in modeling the gas within a radius of 500 (cid:48)(cid:48) as a strongly-warped butotherwise symmetric disk. The disk is made up of rings of material, following near-circularorbits that are concentric but tilted. Because the emission does not peak symmetricallyabout a mean velocity at each point on the sky, we cannot use tasks such as rotcur (Begeman 1987, 1989) to determine the ring orientations by fitting to the mean velocityfield, as measured by the first-moment map. Instead, we built the task inspector in gipsy (Vogelaar & Terlouw 2001) to compare the predictions of such a model to varioustwo-dimensional cuts through the three-dimensional cube of data.Following the convention of Rots (1975) and Begeman (1989), we measure the positionangle p of each gas orbit anti-clockwise from north to the line of nodes (the kinematicmajor axis) on the receding side of the galaxy. Note that this definition of p is 180 ◦ different from that of Schwarz (1985). The orbital inclination i runs from zero as the spinaxis points towards the observer, through 90 ◦ for an edge-on ring, to 180 ◦ .Neglecting the effect of both random motion in the gas and our finite beam size, inspector calculates the expected velocities at which each ring of H i should contribute toa given longitude-velocity cut, or the positions at which its emission should appear in agiven channel map. Fixing the central velocity at 990 km s − gave a slightly better fit thanthe central value of 995 km s − that we derived from the global profile. We placed the ringcenters at the radio continuum source, and adjusted the rotation speeds and the ring anglesinteractively, using inspector to compare model predictions with the position-velocitycuts and channel maps.Figure 9 shows that the position angle of the gas orbits is the best-determined quantity. Inthe top panel, we see that at radii r > (cid:48)(cid:48) the results from inspector are in excellentagreement with those obtained by tracing the ridge line of emission in the extreme velocitychannels. They match fairly well to the model fit by Krips et al. (2005) to the COobservations: see below. Near the galaxy center, the apparent major axis of the gas orbitslies roughly east-west, almost orthogonal to the major axis of the stellar light. It thenturns counterclockwise towards north-south, twisting quite sharply at smaller radii andthen more slowly beyond 300 (cid:48)(cid:48) .The most difficult quantity to determine is the rotation speed. Initially, we used therotation curve fit by Schwarz (1985) to the earlier H i observations. This yielded the model 18 –Fig. 9.— Tilted ring model for the warped H i disk in NGC 3718. In the top panel, redlines and open circles show the position angle p of the receding line of nodes, as definedby the ridge lines of intensity along the three extreme channel maps above and below thesystemic velocity (at 754.9 km s − , 760.1 km s − , 770.5 km s − , 1232.2 km s − , 1227.0 km s − and 1216.6 km s − ). Filled circles and a dashed or solid line show our tilted-ring models inspector inspector et al. (2004). The purple long-dashed line shows thewarped disk of CO from Krips et al. (2005). The second panel shows inclination i , and thethird panel the assumed or fitted circular speed V rot . The bottom panel gives V rot sin i , themaximum speed along the line of sight. The red lines with long and short dashes show theconstraint derived in Section 3.2 above above, and the red star shows the innermost velocitiesseen in CO by Krips et al. (2005). 19 – inspector
1. However, the CO velocities of Krips et al. (2005) show a rise to 250 km s − within 10 (cid:48)(cid:48) of the center. Our 13 (cid:48)(cid:48) synthesized beam is roughly 1 kpc across, so we do notexpect to resolve a rapid central rise in the rotation curve. For the model inspector
2, webegan our iteration with V rot set at 250 km s − , and decreased it in the outer parts onlywhen we could not otherwise obtain a good fit. Figure 7 shows the geometry of thistilted-ring model; the run of position angle and inclination are given in . Figures 10 – 12compare the model predictions with position-velocity cuts and channel maps.The lower panels of Figure 9 show the runs of inclination and rotation speed. Themultiply-peaked velocity profiles illustrated in Figure 8 require that the warped gas diskpasses through edge-on with i = 90 ◦ ; we place this transition between 160 (cid:48)(cid:48) and 220 (cid:48)(cid:48) .Within this radius, lines of sight can pass three times through the disk. The position anglein this region of nearly edge-on gas orbits is 155 ◦ –175 ◦ , running along the dark central dustlane in the left panel of Figure 1. The velocity field of the H i gas is exactly the same for aring of inclination i and one at 180 ◦ − i ; we use the dust distribution in the right panel ofFigure 1 to resolve this ambiguity. There, we see the bright nucleus to the north of thedust lane; so the south side of the dusty gas disk is closest to us. The gas recedes on theeast side, so its spin axis points away from us, meaning that i > ◦ . The warp appearssmooth as the disk twists through edge-on, so we follow Schwarz (1985) in assuming thatthe inclination decreases monotonically to i < ◦ at larger radii.Schwarz (1985) constrained the position angle of the edge-on gas orbit by examining theemission peak closest to the center at velocities close to systemic. He found that thecentroid of that peak moved along a line in P A = − ◦ as the velocity decreased through V sys . This is the behavior expected along an edge-on circular orbit in P A = 157 ◦ . Werepeated this exercise for the present data set as a consistency check, and find that thecentral peak moves along P A = 135 ◦ . Our measured velocity gradient corresponds to aring at radius 90 (cid:48)(cid:48) , where the top panel of Figure 9 shows that the position angle indeedreaches 135 ◦ . Thus the gas orbit at this radius is very close to edge-on.Within 300 (cid:48)(cid:48) of the center the H i gas orbits are less than 10 ◦ from edge-on. So themeasured speed V rot sin i should be very near to the orbital speed itself. Our stipulationthat V rot should decrease monotonically means that our model predicts too high a value for V rot sin i . To avoid this we would need a rotation curve like that of the model inspector R . Closed loops in the channel maps at1190.6 km s − in Figure 3 and at 796.4 km s − in Figure 4 show that either the rotationspeed must drop in the outer disk, or the gas orbits turn closer to face-on. We find thatboth effects are present. At large radii the shape of the total HI map in Figure 6 showsthat the outermost gas orbits turn to i ∼ ◦ , or ∼ ◦ from edge-on. They cannot become 20 –much more face-on, or the predicted east-west extent of gas in channel maps near thesystemic velocity becomes much larger than observed, and the two arms of the “fork” inFigures 10 and 11 are too wide-open. To reproduce the closed loops, we had to reduce themodel rotation speed to about 220 km s − near the outer edge. This behavior is consistentwith the 10%–30% drop in rotation speed that Noordermeer et al. (2007) found to becommon in massive S0 and Sa galaxies, with rotation speeds above 200 km s − .Far from the galaxy center, emission in the channel map near 990 km s − extends almosteast-west for about 200 (cid:48)(cid:48) on each side of the center. However, the fit at this velocity (shownin the top left panel in both Figures 10 and 11) is better on the east (left) side than thewest. Also, Figure 6 shows that the gas furthest to the east and west does not seem to bepart of a complete ring. So we treat our model with caution within 40 (cid:48)(cid:48) and beyond 400 (cid:48)(cid:48) radius.We see in Figure 9 that the molecular gas follows the same warped disk structure as theinner H i layer. Krips et al. (2005) fitted a tilted-ring model to describe the CO kinematics.They chose a model rotation curve close to that of Schwarz (1985): the rotation speed V ( r )is taken as 235 km s − within 100 (cid:48)(cid:48) of the center, rising linearly to 255 km s − at 130 (cid:48)(cid:48) . TheirFigure 12 displays the derived run of tilt angle with radius, relative to a reference planeinclined by 70 ◦ (or 110 ◦ ) to the plane of the sky, and with the approaching line of nodes atPA=-60 ◦ . With respect to that reference plane, their model takes the twist angle toincrease with radius r as twist ∝ cos( tilt ) × r/V ( r ) (compare Equation 1 below). Takingthe reference inclination as i = 110 ◦ and using the tilt and twist angles kindly supplied byDr. Krips, we recovered the inclination and position angles of their model relative to thesky plane, as shown in Figure 9. The position angle agrees well with what we derive fromthe H i observations. The inclination oscillates because of the form that they chose for thetwist, but the product V rot sin i is very close to that for the H i layer.These sets of derived quantities each represent an eyeball fit to a constrained parametricmodel. This contrasts with systematic fitting techniques such as rotcur that are appliedto galaxies with a single-valued velocity field. Because the velocity field represents anintegral over the full data cube, it is smooth and relatively insensitive to the clumpydistribution of emitting gas, and can be compared directly with a model in which gasorbits are uniformly filled. In a galaxy like NGC 3718 we must work with the full 3-D datacube, where the patchy emitting gas lies close to a warped and folded 2-D surface.J´ozsa et al. (2004) present a model for the H i layer in NGC 3718 from TiRiFiC, a newmethod (J´ozsa et al. (cid:48)(cid:48) . Figure 9 shows that therun of position angle is very similar to ours, and the inclination shows the same decreasing 21 –Fig. 10.— Tilted ring model inspector − . Crosses indicate emission from each of themodel rings; the size of the cross increases proportionally with the ring radius. The centralvelocity for each map is given in km s − in the top left corner; maps with the larger labels aredisplaced from the central velocity by the same amount as the maps in corresponding panelsof Figure 11. Other channels are chosen to illustrate features such as the closed contours at1187 km s − . 22 –Fig. 11.— As Figure 10, but for channel maps at velocities below the central velocity. Mapswith the larger velocity labels are displaced from 990 km s − by the same amount as themaps in corresponding panels of Figure 10. 23 –Fig. 12.— Tilted ring model compared with longitude-velocity cuts taken along the east-westdirection. The velocity of the front (closest) portion of each ring is shown by an open bluesquare, and the rear (more distant) portion by a red cross. Large squares enclose symbolsat radius 400 (cid:48)(cid:48) ; large triangles show rings at 300 (cid:48)(cid:48) . Large red and blue circles with inscribedcrosses show rings at 200 (cid:48)(cid:48) ; in the central cut only, these circles almost coincide and areshown as a single light symbol, very close to the central velocity. Rings at 100 (cid:48)(cid:48) appear inthe central cut only, as large circles with central dots. 24 –trend within 400 (cid:48)(cid:48) . The run of inclination differs; J´ozsa et al. (2004) estimate that the gasorbits turn through edge-on closer to the center, at 80 (cid:48)(cid:48) radius and in P A ≈ ◦ , and thatat larger radii the gas remains further from edge-on than indicated by our model-fits. Inthe central 240 (cid:48)(cid:48) the run of V rot sin i falls below the constraint that we derived from thechannel maps of Section 3.2. The implied rotation curve is not monotonic, rising from210 km s − with maxima at 120 (cid:48)(cid:48) and 320 (cid:48)(cid:48) . The differences between the sets of curves inFigure 9 illustrate the difficulties of the fitting methods, the limitations of the data, anddeviations from uniformly filled concentric circular orbits. In Figure 1, we seem to see the stellar disk of NGC 3718 close to face-on. We cannot easilymeasure its orientation from the isophotes, because of the obscuring dust. Fromnear-infrared photometry in the H band (1.6 µ m) within 50 (cid:48)(cid:48) of the center, Peletier &Willner (1993) find isophotes elongated in PA=112 ◦ , with ellipticity (cid:15) ≡ − b/a = 0 .
17 (seetheir Table 4). Tully et al. (1996) give (cid:15) = 0 .
58 at PA=195 ◦ , measured between 150 (cid:48)(cid:48) and250 (cid:48)(cid:48) radius (see their Figure 8 and Table 2), which would correspond to a round disk seen55 ◦ to face-on (assuming an intrinsic axis ratio b/a = 0 . µ m) taken by Tully et al. (1996); only the lower-reslutionimage with 2.052 (cid:48)(cid:48) pixels was used in their paper. Neither shows any sign of the dust lane,even at the center. ¿From their higher-resolution image with 0.753 (cid:48)(cid:48) pixels, we measure anellipticity (cid:15) ≡ − b/a = 0 .
11 to 0.12 at 25 (cid:48)(cid:48) – 27 (cid:48)(cid:48) , which is within the first exponential scalelength of the disk but beyond most of the bulge light (see below). This would correspondto a round disk with intrinsic b/a = 0 . ◦ to face-on. The major axis atPA ≈ ◦ is almost the same as at large radii. However, oval distortions of 10% arecommon in galaxy disks (Rix & Zaritsky 1995; Kormendy & Kennicutt 2004), especiallyamong earlier types (Ryden 2006). In what follows, we assume that the position angle p g where the galaxy disk intersects the plane of the sky lies in the range p g = 195 ◦ ± ◦ .H´eraudeau & Simien (1998) measured velocities along PA=15 ◦ , and find a rise to roughly100 km s − at 20 (cid:48)(cid:48) –30 (cid:48)(cid:48) radius on both sides of the center. According to their Figure 1 thesouthwest side of the stellar disk is receding, just as for the outer H i , so the receding line ofnodes lies near PA=195 ◦ . If the stellar disk is indeed inclined at i =28 ◦ , then for circularspeeds close to 250 km s − we would expect to see motions of about 110 km s − along thekinematic major axis. So these observations are consistent with a round stellar diskinclined with its apparent major axis close to their slit position. H´eraudeau & Simien 25 –(1998) find a central velocity dispersion of 193 km s − (including the factor f bulge in theirTable 1), which drops to about 100 km s − at 30 (cid:48)(cid:48) .From this photometric and kinematic evidence, the galaxy disk appears to be nearlyface-on, as suggested by the dynamical model developed by Sparke (1990) for the warpedand twisted gas layer. However, that model took the plane of the stellar disk to be inclinedwith i g ≈ ◦ at a position angle close to PA=–90 ◦ (see Sparke 2002). Over most of itsradial extent, the H i disk is then tilted by about 80 ◦ with respect to this reference plane,and its twisting could be explained by differential precession. But if the stellar disk indeedhas this orientation, we would expect streaming speeds to be low along the directionPA=15 ◦ explored by H´eraudeau & Simien (1998). It seems more likely that the stellar diskintersects the sky plane along a line closer to PA=15 ◦ (or equivalently PA=195 ◦ ).Accordingly, we abandon the earlier model.The gas orbits of our tilted-ring model nowhere lie close to the plane of the galaxy’s stellardisk. The disk of NGC 3718 seems to be that of an S0 galaxy, substantially free of cool gas.This is similar to NGC 2655 (Sparke et al. ◦ to the stellar disk (van Driel et al. i gas in the system.Although it contains 8 × M (cid:12) of H i gas, with 4 × M (cid:12) of molecular material (Pott et al. × L (cid:12) (Rice et al. et al.
5. Why should the gas layer be warped and twisted?
What might have caused the gas layer in NGC 3718 to become warped and twisted? Adisk of material following orbits tilted away from the galaxy equator will tend to twistbecause of differential precession. In an oblate galaxy, consider a cloud of gas following anorbit tipped by an angle α away from the equator, passing upward through the midplane.The cloud will make a complete vertical oscillation and again cross the midplane travelingupward, before it has made a whole orbit about the center. The tilt of its orbit remains 26 –constant, but the line of nodes, where that orbit crosses the symmetry plane, regresses inthe direction opposite to the orbital motion: (see e.g. Section 5.8 of Goldstein et al. p for an orbit at radius r inclined by an angle α is related tothe circular speed V ( r ) byΩ p = 1 rV ( r ) ∂ (cid:104) Φ (cid:105) ∂ cos α ≡ − (cid:15) Φ cos αV ( r ) /r or − (cid:15) Φ cos α Ω( r ) . (1)Here (cid:104) Φ (cid:105) represents the gravitational potential energy, averaged over the ring ( e.g. Sparke1986), and Ω( r ) = V /r is the orbital angular speed. The quantity (cid:15) Φ measures theflattening of the potential; it is positive for an oblate system, so Ω p is negative. Becausethe orbital periods are shorter towards the center the inner orbits will regress faster, unlessthe galaxy’s flattening increases strongly with radius. Thus a gas disk made up of materialon concentric tilted orbits generally develops a leading twist. Conversely, a disk in aprolate galaxy potential will twist in a sense that trails the rotation.Its very regular velocity field suggests that the outer H i disk of NGC 3718 has been inplace for at least a couple of orbits. For a rotation speed of 230 km s − , the orbital periodat 400 (cid:48)(cid:48) is roughly 900 Myr, implying an age of at least 2 Gyr. Rotation times in the innerdisk are much shorter, and at 40 (cid:48)(cid:48) radius this would correspond to at least 20 orbits. Theposition angle of the gas orbits has twisted by about 120 ◦ between these radii. If that twistrepresents precession in a system of roughly constant flattening (cid:15) Φ , then by Equation 1 wemust have | Ω p | < Ω /
60 for the inner orbits, or (cid:15) Φ cos α < /
60. At first glance, this impliesthat the gravitational potential must be improbably spherical to prevent the disk fromtwisting around itself many times in its ≥ et al. (1992) found a gas disk extending to roughly 7times the radius of the inner edge at 13 (cid:48)(cid:48) (1 kpc at an assumed distance of 15.8 Mpc) thatappears to wrap by almost two complete turns around the galaxy pole. They argue thatthe outer disk is at least six orbits old, corresponding to 40 orbits at the inner edge.Precessional twisting then implies a nearly spherical mass distribution with axis ratio b/a ≥ .
84. The stellar body is flattened with an axis ratio roughly 2:1 ( b/a = 0 . et al. (1982): this had the great advantage of representing a stable equilibriumstate. It requires that the galaxy’s mass distribution is not axisymmetric, but a triaxialspheroid tumbling about its short axis. The potential then supports a family of stable‘anomalous orbits’ described by Heisler et al. (1982), which make up a warped disk. The 27 –dusty gas should settle onto these orbits, in a process studied numerically by Habe &Ikeuchi (1985), Habe & Ikeuchi (1988), Steiman-Cameron & Durisen (1988) and Colley &Sparke (1996). Near the galaxy center, in the core of the gravitational potential, theanomalous orbits circle the long axis. The pole tilts with radius, until at large radii theorbits lie in the ‘equatorial’ plane perpendicular to the short axis, circling it in the oppositesense to the one in which the figure tumbles. The anomalous orbits make up a twisted diskwhich follows a restricted warp: orbits at all radii cross a single line of nodes, which at eachinstant lies along the intermediate axis of the tumbling triaxial galaxy.We can understand the anomalous family as a set of orbits that precess about the shortaxis of the galaxy at exactly the right rate to keep up with the tumbing galaxy potential.Orbits in a triaxial galaxy will precess about the long or short axis at an average rate givenby Equation 1, where (cid:15) Φ is now the average flattening about that axis ( e.g. Steiman-Cameron & Durisen 1984). When the triaxial figure tumbles about its own shortaxis at a rate Ω t , the stable anomalous orbit family consists of just those orbits thatprecess at the rate Ω p = Ω t . For orbits circling the short axis we have (cid:15) Φ >
0, soEquation 1 requires the orbital motion to be retrograde with Ω t <
0. If the system isequally aspherical at all radii and V ( r ) is constant, then Ω p is constant whencos α ∝ r . (2)As Heisler et al. (1982) point out, the anomalous family tilts over to reach the galaxy’sequatorial plane at the radius where the rate Ω p of free precession for an orbit that is onlyslightly tilted from the equator becomes equal to the tumbling speed Ω t .To test whether the anomalous orbit family can represent the warped H i layer ofNGC 3718, we specify the position angle of each gas orbit by the unit vector l along thereceding line of nodes, where the orbit crosses the plane of the sky. The spin axis is along n which is perpendicular to l , and we take m in the plane of the ring to complete theright-handed set l, m, n . We take Cartesian coordinates x, y, z from the galaxy center,with z pointing towards the observer, x to the east and y to the north. In these coordinates,the vectors l, m, n are related to the inclination i and position angle p of Section 4 by l = ( − sin p, cos p, , m = ( − cos i cos p, − cos i sin p, − sin i )and n = ( − sin i cos p, − sin i sin p, cos i ) . (3)The planes defined by two rings with normals along vectors n , n intersect along the line n × n . A restricted warp is one in which this vector points in the same direction for allpairs of rings.We can specify the equatorial plane of the galaxy by the apparent position angle p g andinclination i g of a circular ring lying in that plane. (Note that if the stellar body is triaxial, 28 –the position angle of the galaxy’s apparent major axis may differ from p g .) Defining thecorresponding vectors l g , m g and n g , the angle tilt between a gas ring and that plane isgiven by cos( tilt ) = n · n g = cos i cos i g + sin i sin i g cos( p − p g ) . (4)The ring intersects the galaxy’s equatorial plane along the direction n × n g . We define the twist to be the angle in the equatorial plane between n × n g and the vector l g where theequator intersects the sky plane; sosin( tilt ) cos( twist ) = l g · n × n g = [cos i g sin i cos( p − p g ) − cos i sin i g ] = n · m g , and sin( tilt ) sin( twist ) = m g · n × n g = sin i sin( p − p g ) = − n · l g . (5)With these definitions, tilt = i and twist = p − p g when the galaxy’s equatorial planecoincides with the plane of the sky so that i g = 0. These are related to the angles θ, β ofSchwarz (1985) by tilt = θ and twist = 180 ◦ − β . The pair of angles ( tilt , twist ) describesthe same ring as ( − tilt, twist + 180 ◦ ). Just as for the inclination i , we usually take0 < tilt < ◦ so that sin( tilt ) is positive.The solid curves of Figure 13 show the angles for the H i orbits, relative to a plane with i g =95 ◦ and p g =105 ◦ , close to that of the gas ring at r = 40 (cid:48)(cid:48) , our innermostreliably-determined orbit. As Schwarz (1985) found, over the best-measured portion of thedisk the rings fall close to a restricted warp. (Schwarz’s reference plane corresponds to i g = 104 ◦ , p g = 114 ◦ which is close to the position angle of our gas at 50 (cid:48)(cid:48) radius.) The twistangles of all the rings between 40 (cid:48)(cid:48) and 300 (cid:48)(cid:48) fall within 10 ◦ of a common value. If we takethe ring at 40 (cid:48)(cid:48) to define the polar plane of the galaxy’s potential, then in 40 (cid:48)(cid:48) < r < (cid:48)(cid:48) the orientation of the gas orbits changes almost exactly from polar to equatorial, aspredicted by the model of van Albada et al. (1982).However, there are two difficulties with interpreting the gas motions as material followinganomalous retrograde orbits. First, the outer H i orbits should lie perpendicular to theshort axis of the triaxial potential. Simulations combining dissipative gas with cold darkmatter (Dubinski 1994; Kazantzidis et al. et al. e.g. Kassin et al. i orbits. Also, the shape of the warpdoes not follow the prediction of Equation 2, given by the straight dashed line in thebottom panel of Figure 13. As Schwarz (1985) noted, within 200 (cid:48)(cid:48) of the center the tilt ofthe disk changes too rapidly to fit this description.The stellar disk appears to be tipped by about 28 ◦ from face-on (Section 4 above). So we 29 –Fig. 13.— For the ring model inspector
2, angles with respect to a reference plane atinclination i g and position angle p g . The top panel shows twist measured relative to the ringat 140 (cid:48)(cid:48) . The red solid line refers to the ‘restricted warp’ obtained for i g =95 ◦ and p g =105 ◦ :the twist is nearly constant in the range 100 (cid:48)(cid:48) < r < (cid:48)(cid:48) . The blue line with dots refersto the orientation of the stellar disk implied by the K-band isophotes: i g =28 ◦ , p g =195 ◦ ;the twist is leading. The green dash-dotted line is for i g =152 ◦ and p g =195 ◦ , the otherpossible orientation corresponding to the K-band isophotes. The twist now has a trailingsense. The middle panel shows the angle α between each ring and the galaxy equator inthe corresponding model: α = 90 ◦ − tilt for the restricted warp and α = tilt for the othermodels. The ring inclination decreases monotonically with radius from polar to equatorialfor all the models except the last. The bottom panel shows cos α ; the dashed line shows therelation cos α ∝ r of Equation 2. The run of twist and tilt for the ‘restricted warp’ modeland that with i g =28 ◦ , p g =195 ◦ is given in Table 5. 30 –have either i g =28 ◦ or i g =152 ◦ , depending on whether the east or the west side of the diskis closer to us. The blue curves with dots in Figure 13 shows that for the combination i g =28 ◦ , p g =195 ◦ , the central gas disk is very nearly polar while orbits at larger radius tiltmonotonically towards the galaxy plane. The bottom panel shows that it follows ratherclosely the curve cos α ∝ r that we expect for the anomalous orbit family. However, this isfar from a restricted warp: the twist angle increases by about 120 ◦ between the inner andouter radii. The twist has a leading sense relative to the orbital motion, as we expect fordifferential precession in an oblate galaxy potential.When we choose i g =152 ◦ , the dash-dot line in Figure 13 shows that the tilt is notmonotonic. The gas orbits are nearly polar near the center, then dip by about 20 ◦ , andthen warp up towards the pole and over it at 420 (cid:48)(cid:48) . Because the flattened stellar body ofthe galaxy should dominates the gravitational force within 150 (cid:48)(cid:48) of the center (see below),the potential should be oblate and we expect the precessional twist to have a leading sense.Instead, we see a trailing twist. The shape of the gas layer is neither a stable configuration,nor a natural result of precessional twisting. We do not consider this model further, butadopt i g =28 ◦ for the stellar disk.If the ring is twisted about the galaxy pole, it cannot be in a steady state: e.g. Hunter &Toomre (1969), Sparke (1986), Arnaboldi & Sparke (1994). Instead, the gas orbits sufferdifferential precession according to Equation 1. In the following section, we construct amass model for the galaxy, to examine how fast the gas layer should twist up, and for howlong the warped gas disk might have been in place. This model is similar to those presentedfor Centaurus A by Quillen et al. (1992), Quillen et al. (1993) and Sparke (1996), where thecomplex warped structure results from an interplay between self-gravity and precession.
6. An illustrative dynamical model
We now examine how a tilted gas disk would precess in a simple axisymmetric mass modelfor the disk, bulge and dark halo of NGC 3718. Our model for the stellar component isbased on near-infrared photometry, to minimize the effect of dust absorption. From deepK-band images that trace the galaxy’s light beyond 300 (cid:48)(cid:48) from the center, Tully et al. (1996) measure a scale length h R = 56 . (cid:48)(cid:48) = 4 .
66 kpc. This is longer than the h R = 27 (cid:48)(cid:48) found by Peletier & Willner (1993) in the H band, and by Chitre & Jog (2002) from2MASS K-band photometry, but both of these images were much shallower. Making ourown ellipse fits to the published image of Tully et al. (1996) with 2.052 (cid:48)(cid:48) pixels confirms thelonger scale length; so for our illustrative model we adopt h R = 55 (cid:48)(cid:48) . We follow Tully et al. (1996) in taking the stellar disk to have an intrinsic axis ratio b/a = 0 .
2, and calculate the 31 –forces from this thickened exponential disk as described in Sackett & Sparke (1990).We take the bulge to be spherical. The K-band radial profile in Figure 8 of Tully et al. (1996) appears roughly exponential outside 20 (cid:48)(cid:48) , as does the 2MASS profile measured byChitre & Jog (2002). Since our innermost measured H i orbits lie further out, it does notmatter how we distribute the bulge mass within that radius. For simplicity we model thebulge as a Plummer sphere with core radius r P = 10 (cid:48)(cid:48) .The rotation curve of Figure 9 remains nearly flat to 400 (cid:48)(cid:48) , which is at least four scalelengths of the stellar disk. This, and the high mass-to-light ratio M dyn /L K ∼ (cid:15) = 1 − b/a of the equidensitycontours, the core radius r H and the asympototic circular speed V H . For a given haloflattening, we set r H and V H by requiring that the combined rotation curve V ( r ) from thebulge, disk and halo remains approximately flat. To calculate V ( r ) we use the equatorialrotation curve of the halo from Equation 4 of Sackett & Sparke (1990), but we average theinward pull of the exponential disk over a circular ring at the appropriate tilt angle. Thehalo torque is computed as described in Sackett & Sparke (1990). The torques from thehalo and the flattened disk are added to calculate the precession rate according toEquation 1. We do not include the HI gas mass in calculating the rotation curve: seebelow. Models like that of Sparke (1996) for the warped disk in Centaurus A shows thatthe self-gravity of the warped disk can also affect details of how it resists precessionaltwisting. In this case the disk is very strongly twisted, so this effect is likely to be small,and we do not include it.The top panel of Figure 14 shows the rotation curve from this model. The disk massM d = 5 × M (cid:12) and the halo is chosen to have a small core radius, r H =10 (cid:48)(cid:48) , so therotation speed declines gently with radius. To provide a flat rotation curve at the center,the bulge mass is relatively small, M b = 2 × M (cid:12) ; the overall mass-to-light ratio M/L K = 1 in solar units. This is a ‘maximim disk’ model: the disk and bulge mustdominate the rotation curve within 2 h R to provide the observed declining rotation curve.We initially take the dark halo to be spherical.The middle and bottom panels show the orientation of the gas orbits in our fit inspector2 , relative to a ‘galaxy’ oriented with i g = 28 ◦ and p g = 195 ◦ . As discussed inSection 4 above, the viewing angles i g and p g for the stellar disk are also uncertain. Thepredicted twist is not very sensitive to a change in the inclination i g , but decreasing theposition angle p g slightly will change the sign of cos( tilt ) for the inner, near-polar orbits,and hence the sense of their precession. The open circles show the orientation when wetake p g =175 ◦ . 32 –Fig. 14.— Rotation curve and expected twisting for a dynamical model with M d = 5 × M (cid:12) and M b = 2 × M (cid:12) . The spherical halo has r H =10 (cid:48)(cid:48) and V H =200 km s − . Top: pointsshow the rotation curve of our tilted-ring fit inspector2 in Figure 9; curves show thetotal rotation predicted from the dynamical model (solid), with the contributions of darkhalo (dotted), disk (dashed) and bulge (dash-dot). Middle: angle tilt of the H i orbits from inspector2 relative to a stellar disk with i g =28 ◦ , p g =195 ◦ (dashed line with filled dots) andfor p g =175 ◦ (open circles). An exactly polar orbit has tilt =90 ◦ . Bottom: angle twist for theH i orbits (filled dots for p g = 195 ◦ , open circles for p g =175 ◦ ), and the precessional twistingpredicted by the mass model (line with dots for p g =195 ◦ , line with crosses for p g =175 ◦ ). Thedashed curve shows the result for p g = 195 ◦ , when the halo is flattened with an E3 shape. Alltwists are measured relative to the gas orbit at r = 240 (cid:48)(cid:48) . Measured angles and consequentlythe predicted twists are uncertain within 40 (cid:48)(cid:48) radius. 33 –The bottom panel shows how much twisting a gas disk would suffer over our inferredminimum 2 Gyr lifetime, if it was initially warped but not twisted, so that its tilted orbitsintersected the ‘galaxy’ plane along a single straight line of nodes. Comparing thisprediction to the twist angles derived from our tilted-ring fit, we see that after 2 Gyr themodel would develop roughly the observed pattern of twisting in the region between radius40 (cid:48)(cid:48) and 400 (cid:48)(cid:48) , where our tilted-ring fit is most reliable. Because it warps away from thepole at larger radii, it does not become strongly twisted, as the naive arguments ofSection 5 would suggest.The mass of the H i disk is 16% of that of the stellar disk in our dynamical model. If wehad included it in our rotation curve fit, we would have reduced the mass of both the diskand the dark halo to compensate. The torque from the disk would then be no more than16% less, and precession times would be longer by that same fraction. Our conclusionremains unchanged.Differential precession changes the twist of the gas orbits, but not their tilt with respect tothe stellar disk. Why then should the gas disk have the observed run of tilt? In the middlepanel of Figure 14 we show the run of tilt angle that allows the orbits at all radii to precesstogether in our dynamical model. The curve is for a halo flattened to an E3 shape, butthose for a spherical halo and even an E6 halo lie nearby. The observed run of tilt liesfairly close to this curve. We conclude that the warped gas disk has the shape that it does,because that shape has permitted it to survive far longer than would otherwise be the case.The dynamical model of Figures 14 assigns the maximum plausible mass to the flatteneddisk.
7. Discussion
We have made high-resolution maps of the H i gas in NGC 3718 and its companionNGC 3729. Our data cube for NGC 3718 shows multiply-peaked velocity profiles and acomplex but highly bisymmetric structure. Using inspector , a task in gipsy , we fitted atilted-ring model, in which gas following near-circular orbits about the galaxy center formsa warped and twisted layer. We confirm the conclusions of Schwarz (1985), that theprominent asymmetric dust lane marks the region where the orbits of the (dusty) gas turnedge-on to the line of sight. The molecular gas mapped by Krips et al. (2005) shares themotion of the innermost H i gas. The unusual diffuse spiral arms fall in regions where gasorbits appear to crowd together on the sky. The arms are visible in blue light: new starshave formed in the twisted gas layer. As in other galaxies with extended H i disks (Sancisi et al. i disk can be traced to 500 (cid:48)(cid:48) or 42 kpc from the center. It is fairlysymmetric within 7 (cid:48) or 35 kpc, where the orbital period is roughly a gigayear. So the gasdisk has probably been in place for at least a few orbits at this radius, or 2–3 Gyr. Furtherout, symmetrically-placed spiral-arm fragments to the east and west are visible in both H i gas and blue light. The polar gas disk is still in the process of formation: the eastern armfragment continues as a streamer of gas stretching to a cloud 60 kpc north of the galaxycenter. Sensitive H i maps increasingly reveal such long streamers and tails in the outerparts of disk systems, continuous with the regular velocity field of the galaxy, that mayrepresent gas in the process of joining the galaxy (van der Hulst & Sancisi 2005). However,the gas in polar orbit around NGC 3718 is very dusty; it is not pristine material.NGC 3718 has been classified as a barred Sa galaxy, but this is misleading. The apparentbar is an effect of looking through the edge-on disk of dusty gas, and the peculiar diffusespiral arms instead represent star formation in the warped and tilted gas disk. K-bandphotometry (Tully et al. i gasorbiting near this plane, so the old stellar disk must be almost empty of cool gas. Instead,NGC 3718 is typical of gas-rich early-type galaxies, where H i gas is often found far outsidethe stellar body, and does not share the stellar kinematics (Noordermeer et al. et al. et al. i gasto the most probable plane for the stellar disk, the innermost gas orbits are nearly polar.We do not see gas orbiting in the plane of the stellar disk itself. Thus NGC 3718 is indeeda polar ring galaxy: as in the archetype NGC 4650A ( e.g. Gallagher et al. i disk are nearly polar, the outer orbits tip to lowerinclination. This pattern of tilt minimizes the destructive effects of differential precession,and has allowed the polar structure to survive until the present day. The observed patternof twisting can be explained by a dynamical model for the galaxy in which the gas orbitsprecess freely about the pole of the stellar disk, and the dark halo is roughly spherical.Polar ring galaxies are one of our few tools for studying the three-dimensional shape of thedark halo. Our models for NGC 3718 allows a round dark halo. Steiman-Cameron et al. (1992) obtain a similar result for the twisted dust disk in NGC 4753, concluding that b/a > .
8. The Milky Way’s flattening can be estimated from the near-polar streams ofstars torn from the Sagittarius dwarf galaxy, which undergo differential precession as theyorbit our Galaxy. This process yields confusing results, with some aspects of the streamspointing to a slightly oblate halo ( e.g.
Johnston et al. et al. (2006) for a summary. However, Johnston et al. (2005) 35 –favor the range 0 . < b/a < . b/a < . et al. (1994) deduced that the dark halo of NGC 4650A is considerablyflattened, with 0 . ≤ b/a ≤ .
4, almost as flat as the stellar disk. In the systemA0136-0801, Sackett & Pogge (1995) used Fabry-Perot imaging to map thetwo-dimensional velocity field of the polar ring in H α emission. The kinematic major andminor axes were skewed away from perpendicular, a sign that the gas followed oval orbits.Fitting a dynamical model to the velocity field in conjunction with the spatial distributionof the emitting gas yielded a flattening b/a ≈ . prolate, with axis ratios 0.6–0.7: e.g. Allgood et al. (2006). Adding a baryonic disk flattens the halo in the same sense as the disk(Dubinski 1994), but only to an average axis ratio 0.7–0.8 (Kazantzidis et al. et al. et al. (2004) found that ’aligned-disk’ galaxy mergerscould produce a halo as oblate as b/a ≈ .
5. If the material of the polar ring was a lateaccretion onto the central galaxy, we would expect the halo to be flattened in the samesense as the host’s disk. If the ring gas flowed in along filaments of the ‘cosmic web’, asMacci`o et al. (2006) propose, the dark halo should be oblate close to the host galaxy,becoming prolate and elongated along the filament further out. Bekki (1998) suggestedthat the polar ring represents the disk of a low-surface-brightness galaxy that captured thedense central body by merger. The dark halo might then be aligned with its long axes inthe plane of the ring. Polar rings indeed deviate from the Tully-Fisher relation in the sensethat Bekki’s model would predict (Iodice et al. . < b/a < .
8. But why should we seesuch a pronounced difference among polar ring systems? Perhaps this is simplyobservational selection. Both NGC 4650A and A0136-0801 have ‘classical’ polar rings,lying nearly perpendicular to the host galaxy’s stellar disk. In a galaxy with a flattenedhalo like NGC 4650A, a gas disk tipped far from the perpendicular as that in NGC 3718would rapidly become twisted beyond recognition. Strongly tilted rings would survivepreferentially in systems with the roundest halos. Another possibility is that the halo 36 –shape depends systematically on the galaxy’s luminosity. The Milky Way, NGC 3718 andNGC 4753 are all luminous systems, while NGC 4650A and A0136-0801 are several timesless luminous, with L B ≈ × L (cid:12) .We are very grateful to Elizabeth Wehner and Jay Gallagher for help with optical andnear-infrared images, and especially for Figure 1; to Melanie Krips for supplying details ofher model fit for the molecular gas; and to Marc Verheijen for access to his K-band images.LSS acknowledges support from the National Science Foundation through grantAST-00-98419. GvM and LSS would like to thank the Kapteyn Astronomical Institute ofGroningen University, Netherlands, and the MPI for Astrophysics in Garching, Germanyfor hospitality while part of this work was carried out. We are all grateful to Hugo vanWoerden for his encouragement throughout this project. Finally, we would like to thankour anonymous referee for comments that helped us to improve and shorten the paper.The National Radio Astronomy Observatory is a facility of the National ScienceFoundation operated under cooperative agreement by Associated Universities, Inc. TheWIYN Observatory is a joint facility of the University of Wisconsin-Madison, IndianaUniversity, Yale University, and the National Optical Astronomy Observatories. TheNASA/IPAC Extragalactic Database (NED) is operated by the Jet Propulsion Laboratory,California Institute of Technology, under contract with the National Aeronautics and SpaceAdministration (NASA). This research has made use of NASA’s Astrophysics Data System(ADS). 37 – REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
42 –Table 1. Basic data for NGC 3718 and NGC 3729Parameter NGC 3718 NGC 3729Type a SBa(pec) (T=1) SBa(pec) (T=2)Distance b
17 Mpc 17 MpcCorrected apparent B magnitude c × L (cid:12) . × L (cid:12) Corrected apparent K (cid:48) magnitude c (cid:48) Luminosity 7 × L (cid:12) × L (cid:12) Optical radius R (cid:48) (cid:48) Ellipticity 1 − b/a of stellar disk e i f ◦ ◦ a de Vaucouleurs et al. (1991); numerical type estimated by Tully et al. (1996) b Tully (1998) c Extinction-corrected magnitudes M b,iB and M b,iK (cid:48) from Table 2 of Verhei-jen & Sancisi (2001) d From D in Table 2 of Verheijen & Sancisi (2001) e For NGC 3718, from Section 3.3; for NGC 3729, from Table 4 of Ver-heijen & Sancisi (2001) f We assume a circular disk with intrinsic axis ratio
B/A = 0 .
2, so thatcos i = [( b/a ) − ( B/A ) ] / [1 − ( B/A ) ] 43 –Table 2. Parameters of H i ObservationsDates of observation 1992.25R.A. pointing center (J2000.0) 11 h m s δ pointing center (J2000.0) +53 ◦ (cid:48) (cid:48)(cid:48) Synthesized beam 13 . (cid:48)(cid:48) × . (cid:48)(cid:48) Effective Velocity coverage – km s −
708 – 1253Number of channels 106Velocity resolution – km s − a Noise – mJy/beam (rms) 0.39 / 0.30 a Noise – K (rms) 1.43 / 1.10 aa For the high- and low- velocity resolution cubes, re-spectively.Table 3. 20 cm Continuum EmissionParameter NGC 3718 NGC 3729Right Ascension (2000) 11 h m s h m s Declination δ (2000) +53 ◦ (cid:48) (cid:48)(cid:48) +53 ◦ (cid:48) (cid:48)(cid:48) Flux at 20 cm (mJy) 14 . ± . ± i parameters of NGC 3718 and NGC 3729Parameter Unit NGC 3718 NGC 3729Systemic velocity V sys a km s −
995 1063H i full width W km s −
476 242H i flux integral c Jy km s − i radius 8.3 (cid:48) = 41 kpc 1.4 (cid:48) = 5 kpcH i mass d M (cid:12) e M (cid:12)
500 35M(H i )/L B M (cid:12) /L (cid:12) a The velocity symmetric with respect to 20% of the peak valueon the profile wings. b Full width at 20% of peak flux as described in the text. c The total area under the global profile. d Calculated from the total flux assuming an optically thinmedium. e For NGC 3729, adopting the inclination i from Table 1. 45 –Table 5. For the model inspector
2, position angle and inclination derived for the gasorbits, and their tilt and twist angles relative to two reference planes: (a) that of a‘restricted warp’ and (b) that indicated by the K-band isophotes of the inner galaxyRadius PA i tilt (a) twist (a) tilt (b) twist (b)arcsec deg deg i g = 95 ◦ p g = 105 ◦ i g = 28 ◦ p g = 195 ◦◦